Numerous calculation in geodesy require calculation of mean topographical effects over grids of cells. Such calculations are performed by dividing each cell into an array of points, computing the topographical effect at each point, and then averaging the point effects the mean effect over that cell. For cells that are 1 arc-minute in size, mean topographical effect estimates may differ by tens of milligals when calculated with different spacings of computation points in the array. In areas of rough topography, a point spacing of 3”, consistent with the resolution of widely available global DEMs, is necessary to ensure milligal level accuracy, but the corresponding computation for 400 points per 1’ cell requires substantial computation time. We propose an alternative approach whereby averages for each cell are instead calculated at resolutions of 30”, 10”, and 6”. The averages converge systematically in all cases according to a reciprocal function, and by fitting a reciprocal function to the 30”, 10”, and 6” results, the value for a 3” spacing can be computed to better than 1 mGal accuracy about 80% of the time even in rough topography. Furthermore, the asymptote of the reciprocal function provides an estimate of the mean value for differentially small sub-cells that likely exceeds the accuracy of the 3” sub-cell computation. Errors in the produced averages may also be estimated based on the fit of the reciprocal function to the 30”, 10”, and 6” results.